Incompleteness
Page 25
reputation of, 13–14, 19–20, 35, 38, 221n
as scientific realist, 41–43, 45, 254–56
as U.S. citizen, 233
Eisenhower, Dwight D., 30
“either-or” propositions, 202–5
Elements (Euclid), 130n
Emperor’s New Mind, The (Penrose), 25, 201
empiricism, 32, 74–80, 84–90, 107, 111, 137, 215, 258–59, 267n
Encyclopedia of Philosophy, 23–24, 26
Epimenides, 166
epistemology, 17n, 24–25, 27, 29, 121–22, 137, 154n
Equivalents of the Axiom of Choice (Rubin and Rubin), 224n
Euclid, 128–29, 137, 185–86, 225n
evolution, 31–32, 199
excluded middle, law of the, 143n
existentialism, 135–36, 189
Fackel, Die, 71
Feferman, Solomon, 265n
Feigl, Herbert, 84–86, 105–6, 109, 110, 136–37, 160, 213–14, 228
Fermat, Pierre de, 155
Fermat’s last theorem, 155n
Fields Medal, 47n
finitary proofs, 118, 144, 148, 154n, 163, 184n, 185–88, 212
Flexner, Abraham, 14–20, 33, 219, 229, 237, 245
formalism, 26, 46n, 86–88, 98–102, 128–29, 132, 134–45, 148, 149, 161
Forman, Philip, 233–34
Franz Joseph, Emperor of Austria, 69
Frayn, Michael, 37–38
Frege, Gottlob, 56n, 81, 83, 91–92, 96n, 111, 119, 128–29, 144, 218
Freud, Sigmund, 70, 71
From A Logical Point of View (Quine), 214
From Mathematics to Philosophy (Wang), 111
Fuld, Mrs. Felix, 14–15, 16, 237
Furtwängler, Phillip, 58, 61, 115n, 204n, 222n
galaxies, rotation of, 258–59
games, 122, 134, 136, 141
game theory, 33n, 40, 100–101
Gauss, Carl Friedrich, 130n
Gedanken-experiments (thought-experiments), 65, 66
Geertz, Clifford, 242–43
Gentzen, Gerhard, 184n
geometry, 62, 85, 123–29, 130n, 131, 137–38, 141–42, 225n
God, existence of, 77–79, 209–10
Gödel, Adele Nimbursky (Porkert), 33, 208, 209, 223–26, 228–29, 230, 248, 249, 250
Gödel, Escher, Bach: An Eternal Golden Braid (Hofstadter), 26
Gödel, Kurt:
ambition of, 29, 47, 51, 60, 64, 118–19, 187n, 194, 222, 259–61
authority respected by, 236–40, 244–45
author’s encounter with, 211–12, 241
birth of, 53
childhood of, 53–58, 260–61
correspondence of, 33, 60, 61, 111–12, 116–17, 192, 227, 234, 235–36, 252n, 266n
death of, 250–52, 260–61
Dozent and Privatdozent appointments of, 156, 220–22, 226–28, 251n
education of, 29, 33, 35, 51, 53, 57–59
German background of, 53–54, 57–58, 124–26
Gibbs lecture of (1951), 202–3
Harvard lectures prepared by, 215
heart ailment of, 56, 229, 260
honorary degrees of, 222–23, 231–32
ill health of, 56, 220, 229, 248–50, 260
influences on, 58–59, 111–12, 115–16, 193
at Institute for Advanced Study, 13–14, 20–23, 30–34, 48, 207, 211–12, 215, 219, 222–25, 229–31, 234–50
intellectual isolation of, 30–32, 35–38, 44–48, 60–61, 213–19, 236–50, 259–60
“interesting axiom” of, 20–21, 30–31, 48, 55, 236
Jewish identity misattributed to, 116, 219–20, 226–30
at Königsberg conference (1930), 65–67, 147–49, 150, 155–62, 195n, 201, 219
language as viewed by, 110, 158, 261
legalistic tendencies of, 232–34, 250–51, 252
legends about, 30–32, 208–13, 232–33, 258–59
literary remains (Nachlass) of, 59–62, 89, 115–17, 213, 232n
logical positivism opposed by, 87–88, 109, 213–19, 236, 266n
as logician, 20–21, 24, 29–30, 31, 48–51, 55, 57, 58–59, 64–65, 74, 76n, 83, 116, 138, 205, 210, 219, 232–34, 236, 239–40, 251–52, 259
marriage of, 33, 208, 209, 223–26, 228–29, 230, 248, 249, 250
mathematics revolutionized by, 21–23, 38–40, 59, 64, 161, 207, 218–23, 229–32
memorial service for, 240, 251–52
paranoia of, 30, 48–49, 56–57, 204–5, 211, 224, 229, 240–41, 246–50
personality of, 30–32, 48–49, 56–57, 59, 61, 75–76, 114–15, 223–31, 265n
Ph.D. dissertations of, 147, 150, 153–54, 156, 159, 186, 221–22
photographs of, 22, 55, 253
physical appearance of, 14, 211, 228–29
physics as interest of, 34, 35, 58, 213n, 253–61
as Platonist, 44–48, 59, 61–64, 73, 75, 87–88, 104n, 110–13, 116–17, 154, 185, 192, 194, 213–19, 244, 256, 260
as political exile, 13–14, 34–35, 53, 58, 124–26, 219–22, 225–34
proof of, see incompleteness theorems
publications of, 34n, 60, 111–12, 212–13, 216–18, 246, 257, 265n
questionnaire answered by, 60–61, 115–16, 193
reputation of, 22–23, 35, 67–68, 147, 207–15, 220–23, 229–31, 248, 251–52, 259, 272n
reticence of, 57–58, 59, 61, 75–76, 109, 110, 114–15, 116, 135–36, 149, 156–58, 194–95, 207–15
rheumatic fever, 56, 229
self-starvation of, 211, 248–50
at University of Vienna, 29, 33, 35, 51, 53, 58–59, 67–102, 115n, 147, 150, 153–54, 156, 219–22, 225–30, 251n, 255, 260
as U.S. citizen, 54, 232–34
in Vienna Circle, 108–13, 115, 116, 117, 135–36, 160–61, 193, 213–14, 216, 228, 236, 266n
Yale lectures of (1941), 212
Gödel, Marianne, 33, 53, 54, 55, 192, 223–24, 230, 234, 235–36, 252n
Gödel, Rudolf (brother), 54, 55, 56
Gödel, Rudolf (father), 53–54, 55, 223
Gödel numbering, 67, 156, 162, 165n, 167, 169–77, 179
Gödel’s Proof (Nagel and Newman), 156
“Gödel’s Theorem” (Encyclopedia of Philosophy), 23–24
Goethe, Johann Wolfgang von, 58
Goldbach, Christian, 87n, 155n, 217
Goldbach’s conjecture, 155n
Gomperz, Heinrich, 59, 61, 73, 115n
Gomperz, Theodore, 59
Grandjean, Burke D., 60–61, 111–12, 115–16, 193
Grundgesetze der Arithmetic (Frege), 91–92
Grundlagen der Geometrie (Hilbert), 137–38
Hahn, Hans, 82–84, 97, 116, 157, 159, 221, 228
Hahn-Banach extension theorem, 82
Hapsburg Empire, 53–54, 68, 69–70, 71
Hardy, G. H., 46–47
Heidegger, Martin, 39
Heisenberg, Werner, 21–22, 36–38, 39, 41–42, 43, 51, 214
Hempel, Carl, 88–89, 214
Herzl, Theodore, 70
Heyting, Arend, 148
Hilbert, David, 46n, 49, 119–20, 123–24, 136–45, 148, 153, 161, 162–63, 164, 184n, 185, 187, 188, 197n, 198, 212, 218
Hintikka, Jaakko, 65–66, 99, 147, 267n
Hirshfeld, Leon, 226–27
“History of the Gödel Family” (R. Gödel), 54
History of Western Philosophy (Russell), 102n
Hitler, Adolf, 13, 225, 227
Hofstadter, Douglas, 26
Hörmann, Theodor, 70n
Hume, David, 77, 267n
incompleteness theorems, 155–88
author’s explication of, 164–88
completeness vs. consistency in, 23–24, 66–67, 140–44, 153–56, 158, 160, 162, 167–68, 169, 175–76, 181–88, 192, 196n–97n, 198, 221
as conceptual models, 25–26
development of, 23, 35, 47, 59, 64–67, 74, 155–61, 260–61
diagonal lemma in, 179–82
digit juxtaposition in, 172–7
4
first theorem of, 23, 24, 35, 39, 50, 65, 74–75, 134–35, 158, 162, 164–83, 197n, 202, 211, 241, 270n
formalism and, 134–35, 145, 149, 161–64, 167, 168–69, 175–77, 178, 182–88, 198, 201–2, 204, 218
Gödel numbering in, 67, 156, 162, 165n, 167, 169–77, 179
logical positivism contradicted by, 74–75, 99, 104n, 112–13, 160, 161, 188–89, 191, 266n
as “logische Kunststücke,” 117, 118–19, 189, 268n–69n
mathematical implications of, 24–25, 28, 51, 74–75, 158–64, 187–88, 190, 221–23, 231–32, 251–52, 260–61
mechanical procedures in, 169–77, 183
metamathematical implications of, 24–29, 35–39, 50–51, 99, 104n, 118–20, 163, 169–77, 187–94, 198–205, 222
musical analogy for, 156, 165, 176–77
number properties in, 172–74, 178–83, 184, 187
objectivity and, 38–40, 50–51
paradoxes and, 49–51, 65, 66–67, 99, 164, 165n–66n, 179n, 184–85
philosophical implications of, 27–29, 67, 191–92, 198–205, 211
popular misinterpretation of, 23, 24–26, 37–40, 61, 74–76, 158, 218, 266n
preliminary definitions for, 164–65
proof of, 66–67, 74–75, 115, 117–18, 153–54, 158, 162–64, 167–68, 170–71, 175–83, 184, 187–88, 191, 196n–97n
publication of (1931), 24, 165n, 195n, 197n
public disclosure of, 66, 74–75, 110, 147–49, 150, 155–62, 195n, 201, 219
relativity theory compared with, 36–40
scientific impact of, 21–23, 161–62, 187–88, 194–205, 231–32
second theorem of, 23–24, 35n, 74–75, 158, 162–63, 183–88, 212
statement of, 23–24, 26, 74–75, 155–56
syntactic relationships in, 171–72, 175–77, 181–83, 188
Turing’s interpretation of, 190, 195–97
Viennese cultural context of, 67–102
von Neumann’s reaction to, 161–63, 187
well-formed formulas in (wffs), 168–75, 178, 181
Wittgenstein’s reaction to, 89–90, 99, 100, 102, 111–20, 158, 188–95, 218, 268n–69n
infinity, 143, 144, 148, 154n, 185–86, 198, 224n–25n
Institute for Advanced Study:
appointments to, 229n, 236–45
directors of, 14–20, 33, 211, 219, 229, 237–39, 241–45
Einstein as member of, 13–14, 19–20, 48, 234–36, 237
founding of, 14–16
Gödel as member of, 13–14, 20–23, 30–34, 48, 207, 211–12, 215, 219, 222–25, 229–31, 234–50
mathematics faculty of, 16–20, 31, 161–62, 195, 229n, 236–45
political exiles at, 13–14, 34–35
Introduction to Mathematical Philosophy (Russell), 117
intuition, 17n, 40, 67, 100, 121–34, 143–44, 148, 188, 198, 202, 203–4, 216–17, 252, 254n
intuitionism, 143–44, 148, 252
“Intuitionist Foundations of Mathematics, The” (Heyting), 148
Irrational Man: A Study in Existentialist Philosophy (Barrett), 38–39
“Is Mathematics Syntax of Language?” (Gödel), 265n
isosceles triangles, 124
Jews, 19, 70, 79n–80n, 97, 116, 126, 219–20, 226–30
Kafka, Franz, 170, 252
Kant, Immanuel, 73, 185, 213n, 267n
Kaysen, Karl, 211, 242–45
Kesten, Herman, 68
Khowâsizm, Abu Ja’far Mohammad ibn Mûsâ al-, 132n
K.-K. Staatsrealgymnasium mit deutscher Unterrichtssprache, 57–58
Kleene, Stephen, 251
Klepetař, Harry, 58
Kline, Stephen, 195n
knotted curves, 238–39
Kochen, Simon, 24n, 170, 187n, 231, 240, 241, 251–52
Kraus, Karl, 68, 71–72
Kreisel, Georg, 119
Kuhn, Thomas, 161, 194
language:
abstraction and, 71–72, 78–79, 158
“games” of, 40, 100–101
meaning and, 78–80, 86–87, 110, 150–54, 158, 261
paradoxes in, 50–51, 102–3, 189
syntax of, 86, 87, 100, 103, 107–8, 112, 189–90
truth and, 98–99, 104, 189–90
Language, Truth, and Logic (Ayer), 89, 214
Leibniz, Gottfried Wilhelm, 27n, 32, 48, 61, 137, 209, 236, 247
Leoncavallo, Ruggero, 176–77
Lewis, Clarence I., 82
“liar’s paradox,” 49–50, 165n, 166, 179n, 196n
light, properties of, 35, 45, 85
Lobachevsky, Nicolai Ivanovich, 130n
logic:
abstract nature of, 38, 71–72, 78–79, 117, 154n, 158, 192, 198–205, 260–61
contradiction in, 92, 93n, 101–2, 118, 133–34, 135, 140–42, 192, 196n
deduction in, 40, 55n–56n, 86, 136, 142, 168–69
formal, 24, 90–93, 112–13, 114, 189–90, 195–96, 241, 267n–68n
“limpid,” 150–54, 158, 221, 222
paradoxes in, 49–51, 65, 66–67, 90–93, 99, 100, 102–3, 117–18, 142–43, 145, 164, 165n–66n, 179n, 184–85, 189, 192, 196n, 198
predicates in, 150–54, 158
proofs in, 144n, 196n
propositions in, 44, 78–79, 85–86, 90–93, 97–103, 150–54, 268n–69n
rules of, 100–102, 153–54
syllogistic, 56n
symbolic, 82, 172–74, 267n–69n
syntactic nature of, 86, 87, 100–101, 103, 107–8, 112, 150–54, 160, 161, 163, 171–72, 175–77, 181–83, 188, 189–90
tautologies in, 86, 98–99, 106, 123, 148, 150, 151
truth in, 44–45, 49–51, 98–102, 116–17, 152–54, 196n, 268n–69n
variables in, 151–52, 173
logical positivism, 43–44, 74–80, 84–90, 97–109, 112–13, 119, 160, 161, 188–89, 191, 213–19, 236, 266n
“Logical Positivism: A New Movement in European Philosophy” (Blumberg and Feigl), 84–85
Loos, Adolph, 70, 72
Lucas, John, 200–201, 211
Mach, Ernst, 80, 85, 112
“Main Ideas of Logicism, The” (Carnap), 148
Man Without Qualities, The, (Musil), 57n
Marchet, A., 228
Marx, Groucho, 27
“Mathematical Problems” (Hilbert), 138–39
Mathematician’s Apology, A (Hardy), 46–47
mathematics:
a priori nature of, 17–18, 27, 28, 48, 50n, 76, 85–86, 108, 121–22, 133, 134, 136, 260
axioms of, 75, 112n, 122–34, 140–41, 144, 163, 185, 186–88, 200, 216–17, 224, 231
descriptive content in, 86, 87–88, 111, 136, 140–41, 187–88
diagrams and sketches for, 122–24, 125
formalism in, 26, 46n, 86–88, 98–102, 128–29, 132, 134–45, 148, 149, 161
Gödel’s influence on, 21–23, 38–40, 59, 64, 161, 207, 218–23, 229–32
meta-, 24–29, 33, 35–40, 44–48, 50–51, 62, 64–65, 67, 99, 104n, 118–20, 134, 154n, 156, 163, 169–77, 187–94, 198–205, 222
objective reality in, 27–28, 35–48, 111–12
philosophy and, 27–28, 59, 61–64, 100–102, 194
physics compared with, 29–30, 35, 36–37
proof in, 86–88, 121–22, 136–37, 138, 140–42, 143n, 144–45, 148–49, 155–59, 243n, 251, 260
as pure science, 16–17, 26–27, 85–86, 99–100
reality of, 44–48, 64, 121–22, 134–35, 140–41
rules of, 101–2, 121–23, 126–33, 136, 137, 154n, 170, 198, 200, 203
symbols of, 86–87, 99–100, 123–24, 131–32, 133, 136–37, 154n, 167, 198, 267n
syntactic nature of, 100–101, 112, 131–32, 188, 198–99, 267n–68n
theorems in, 118, 126, 127–28, 131, 134, 135, 140–41, 200
truth of, 86–87, 98–102, 107–11, 137–38, 140–41, 148–49, 158–59, 188, 202–5
see also arithmetic; geometry; logic
&nb
sp; Mauthner, Fritz, 103
Mendel, Gregor, 53n
Menger, Karl, 84, 110, 117–18, 149, 158, 220, 225–26, 227, 228, 245–46, 247, 250–51, 266n
metaphysics, 76, 85, 104n, 110–11, 112, 216
Milnor, John, 238–39
mind, see cognition
Misner, Charles, 258
model theory, 187n, 194, 251–52
modus ponens, 92n
molecular theory, 80n–81n
Monk, Ray, 149
Montgomery, Deane, 238, 239–40
Moore, G. E., 97, 267n
Morgenstern, Oskar, 32–33, 48, 211, 224, 230, 232–34, 246, 247–51
Morrell, Ottoline, 94, 113
Musil, Robert, 57n
mutation, genetic, 32n
mysticism, 83–84, 104n, 106, 107, 191–92
Nagel, Ernest, 156
Nagel, Thomas, 31–32
Natkin, Marcel, 109
“Nature of Mathematics, The: Wittgenstein’s Standpoint” (Waismann), 148–49
Nazism, 13, 19, 70, 79, 124–26, 212, 225–30, 251n
Nelböck, Johann (Hans), 79n
Neurath, Olga, 82
Neurath, Otto, 81–84, 110
Newman, James R., 156
Newton, Isaac, 58, 254n
New York Times, 241, 243, 254n
“Nicolas Bourbaki,” 243n
Nietzsche, Friedrich, 39
non-Euclidean geometry, 129, 130n, 225n
non-finitary proofs, 154n, 212
nonstandard analysis, 241
numbers:
even, 155n, 165n
natural, 45, 112n, 127, 139, 163, 165n–66n, 172–74, 178–83, 187
prime, 46–47, 87, 155n, 172, 176, 217
properties of, 172–74, 178–83, 184, 187
real, 112n, 139
sets of, 62, 83, 90–93, 112n, 129–30, 131, 135, 139, 142, 144–45, 212, 216–17, 224–25, 252
theory of, 23, 58–59, 61, 64, 83, 85–86, 90–93, 118, 131, 135, 155n
“On computable numbers, with an application to the Entscheidungsproblem” (Turing), 197n
ontology, 76, 209–10
Oppenheimer, J. Robert, 237–39, 242, 245
Organon (Aristotle), 55n–56n
Ovid, 209
Pagliacci, I (Leoncavallo), 176–77
parallels postulate, 129n–30n, 136, 185–86, 225n
Pascal, Fania, 114
Peano, Giuseppe, 96, 127, 128
Peebles, Jim, 258–59
Penrose, Roger, 25, 201–2
philosophy, 27–29, 34, 59, 61–64, 67, 100–102, 103, 191–92, 194, 198–205, 211
Philosophy of Mathematics (Benacerraf and Putnam, eds.), 111–12, 216–18
physics, 28–30, 33, 34, 35–48, 51, 58, 65, 213n, 253–61, see also relativity theory
Pinker, Steven, 32n
Planck, Max, 80
Plato, 39, 44, 61–64, 113, 121, 137, 199, 255, 260